When talking about fibers in topology and geometry, especially in fiber bundle, vector bundle and fibration, it is said that they are discrete spaces. But in some diagrams I see, they are often depicted as real lines over base space, which seems to suggest they are not discrete spaces. From what I understand intuitively, discrete spaces are the spaces where all points are isolated, and a real line is definitely not a discrete space. So, are depictions just misleading? Or am I understanding something wrongly..?
OK, so here's the reference to what I am talking about (http://en.wikipedia.org/wiki/Covering_space#Formal_definition):
Let X be a topological space. A covering space of X is a space C together with a continuous surjective map
$p \colon C \to X\,$
such that for every x ∈ X, there exists an open neighborhood U of x, such that p−1(U) (the inverse image of U under p) is a union of disjoint open sets in C, each of which is mapped homeomorphically onto U by p.
The map p is called the covering map, the space X is often called the base space of the covering, and the space C is called the total space of the covering. For any point x in the base the inverse image of x in C is necessarily a discrete space called the fiber over x.
A fiber bundle with discrete fiber is precisely a covering space. Since the definition you supplied is only talking about covering spaces, there is no contradiction here.
In general, the fiber of a fiber bundle can be any topological space $F$. Just consider the trivial fiber bundle $X \times F$ over any space $X$.